Skip introductory remarks and go directly to:

1) view video of a recent presentation (Perimeter Institute, Waterloo, Canada, October 1, 2009) 

2) read an introductory outline of what we meant by Quantum Thermodynamics in the 1980's

3) browse a list of references and download their pdf's


Introductory remarks


During the 20th century, the laws of mechanics have been profoundly modified by two major revolutions in our understanding of natural sciences: quantum theory and relativity. The laws of thermodynamics, instead, have survived both revolutions, unaltered and strengthened. The deepness and generality of the principles of thermodynamics, and the importance and conceptual difficulty of its enigmas have enflamed the minds of all principal developers of modern physics. For these reasons thermodynamics is certainly not a dead subject, but one of the most lively scientific disciplines: a perpetual generator of scientific thinking and technological progress.


While the importance and the empirical successes of the laws of thermodynamics and their applications have never been questioned, their profound physical significance and domain of validity have been and still are at the center of several debates and controversies among the different schools of thought.


Joseph Henry Keenan (1900-1977) has been a major contributor to the field of thermodynamics, both from the applications point of view (he studied the properties of steam and coauthored the most authoritative steam tables used by generations of engineers) and from the theoretical point of view. In his 1941 book on Thermodynamics he introduced the concept of availability, laying the foundations of today’s widely applied concepts of second law efficiency and exergy analysis. This book has had an authoritative and continuous influence on teachers of thermodynamics, in all branches of engineering, and throughout the world. Through his writing and teaching, Professor Keenan brought to the engineering profession the fundamental work of J. Willard Gibbs in thermodynamics, which, for the most part, had been overlooked by engineers and scientists for five decades. The initial motivation for this development was the allocation of fuel costs in a process with many outputs. The concepts of availability soon became widely used in chemical engineering and power-plant engineering, particularly abroad. In the United States, it has in a sense been tardily rediscovered and has recently become an important tool in the shaping of a national energy policy. See the website of the International Symposium “Meeting the Entropy Challenge” held at MIT, October 4-5, 2007 in honor and memory of Prof. Keenan (the website contains a link to the full video recordings of the event). During the 2007 Fall term, prof. Beretta visited the Mechanical Engineering Department at MIT and gave a monographic 12 credit course on Quantum Thermodynamics, 2.997 Advanced topics in Mechanical Engineering (click here to see the outline of the course), during the 2008 Fall term, the same course was given at Northeastern University (course MTM G374 Quantum Thermodynamics). These courses are the outgrowth of the course 2.452J Quantum Thermodynamics jointly given by proff. Beretta and Gyftopoulos in 1983-4-5-6 at MIT, which were among the first courses* with such title. Later on, the same title over the years has been adopted with a variety of different meanings, generally associated with different theories of nonequilibrium.

In the late 1950's and the 1960's, Professor Keenan contributed with George N. Hatsopoulos to a fundamental reinterpretation of thermodynamics that is applicable to a much wider range of systems, states, and physical phenomena than any other interpretation presented in the past. The book “Principles of General Thermodynamics” published in 1965 by Wiley is another milestone in the history of thermodynamics, which contains the most innovative formulation of the second law after the work of J.W. Gibbs and as such has had a great influence on all subsequent books on thermodynamics, because of the depth and generality of the conceptual development, and of the novel general formulation of the second law in terms of existence and uniqueness of stable equilibrium states, that has come to be known as “the Hatsopoulos-Keenan statement of the second law”. 

George Hatsopoulos deserves to be considered a "true pioneer" of the foundations of both classical and quantum thermodynamics. His 1965 book with Joe Keenan is a milestone contribution to understand the role of stability of equilibrium in the second law. The four papers he wrote with Gyftopoulos in 1976 are the first attempt to establish a coherent theory of quantum thermodynamics. In the Spring of 1971, George offered at MIT the first course ever taught worldwide entitled 'Quantum Thermodynamics'. He was so ahead of his times, that it took another twentyfive years before interest in the physics community began to sprout and blossom, and another ten to explode to the current viral level. 

*Read here the list and official outlines of the courses taught at MIT between 1966 and 2007 by Hatsopoulos, Gyftopoulos, and Beretta on 

Read here a brief personal memory by Gian Paolo Beretta prepared for the January 14, 2021 event "George Hatsopoulos - Celebration of Life and Philantropy at MIT" held by the MIT Department of Mechanical Engineering.

Hatsopoulos and Keenan formulated a number of seeding questions about the general foundations of thermodynamics, that in the 1970's and 1980's motivated two lines of research at MIT. The first has been the reformulation of the axiomatic foundations of thermodynamics based on rigorous and unambiguous definitions, and a rearranged order of exposition so as to remove the logical inconsistencies and limitations typical of traditional expositions. This new approach, pioneered by Hatsopoulos and Elias P. Gyftopoulos,  and completed by Gyftopoulos and Gian Paolo Beretta served as the basis for teaching graduate thermodynamics courses at MIT for the last three decades. It is thoroughly covered by the book “Thermodynamics. Foundations and Applications” by Gyftopoulos and Beretta published in 1991 by MacMillan and republished by Dover in 2005. The second line of research, tackled the long standing apparent conflict between the postulates of thermodynamics, including irreversibility, and those of quantum mechanics, and identified a plausible resolution in postulating that the ordinary postulates of quantum theory must be complemented by the Hatsopoulos-Keenan statement of the second law, for all sistems, including a single particle. The resulting unified theory of mechanics and thermodynamics, was pioneered by Hatsopoulos and Gyftopoulos, and completed by James L. Park and Beretta, who addressed the problem of describing irreversible processes within the unified theory. 

Given that many ideas about thermodynamics were introduced in the 19th and early 20th centuries, one might assume that there is little controversy about its foundations and applications. However, even a cursory review of the relevant literature shows that this is not the case.  The ideas of thermodynamics have been the subject of controversy ever since their inception, controversy that continues even today. Though dated, the following comments continue to be valid. Obert (1960) writes: Most teachers will agree that the subject of engineering thermodynamics is confusing to the student despite the simplicity of the usual undergraduate presentation. Again, Tisza (1970) states: The motivation for choosing a point of departure for a derivation is evidently subject to more ambiguity than the technicalities of the derivation ... In contrast to errors in experimental and mathematical techniques, awkward and incorrect points of departure have a chance to survive for a long time. Werhl (1978) writes: It is paradoxical that although entropy is one of the most important quantities of physics, its main properties are rarely listed in the usual textbooks on statistical mechanics.  Lindblad (1983) gives a large number of different expressions for entropy and comments: The entropy function is not unique.  Instead there is a family of such functions, one for each set of thermodynamic processes allowed by the experimenter's control of the dynamics of the system through the external fields.  This scheme is in line with the philosophy described by Jaynes' dictum (1957): Entropy is a property, not of the physical system, but of the particular experiments you or I choose to perform on it. Truesdell (1986) identifies several different statements of the second law.  Bunge (1986) lists about twenty ostensibly in-equivalent but equally vague formulations of the second law. Mehra and Sudarshan (1972) among many other very important recommendations declare: We maintain that for the explanation of statistical mechanical phenomena the law of evolution is not Hamiltonian, and by creating a generalized dynamics which is essentially non-Hamiltonian we can rid ourselves of all ad-hoc intermediate assumptions.  Thereby we can also shed all the paradoxes that arise in connection with Boltzmann's equation and the H-theorem, as well as the pretense of the mechanical explanation of the second law of thermodynamics

It is noteworthy that the recommendation just cited was made also by Sadi Carnot (1824) about a century and a half earlier in his pioneering and trail blazing Reflections on the motive power of fire.  He said: In order to consider in the most general way the principle of the production of motion by heat, it must be considered independently of any mechanism or any particular agent.  It is necessary to establish principles applicable not only to steam engines but to all imaginable heat-engines, whatever the working substance and whatever the method by which it is operated. And then Carnot continues: Machines which do not receive their motion from heat ... can be studied even to their smallest details by the mechanical theory.  All cases are foreseen, all imaginable movements are referred to these general principles ... . This is the character of a complete theory.  A similar theory is evidently needed for heat-engines.  We shall have it only when the laws of physics shall be extended enough, generalized enough, to make known before hand all the effects of heat acting in a determined manner on any body. 


The growing interest during the last several decades in quantum dynamical models of systems undergoing irreversible processes has been motivated by impressive technological advances in the manipulation of smaller and smaller systems, from the micrometer scale to the nanometer scale, and down to the single atom and single photon scale. The laws of thermodynamics, that fifty years ago were invariably understood as pertaining only to macroscopic phenomena, have gradually earned more attention and a central role in studies of mesoscopic phenomena first, and of micro-, nanoscopic and single particle phenomena more recently. Physicists are now studying thermal machines in which the working fluid is a single photon or a single electron constrained within a subset of electronic levels in a single atom.  The pioneering ansatz, first introduced by Hatsopoulos and Gyftopoulos in 1976 (references can be downloaded below), that thermodynamics applies as well to the single particle level, seems close to being vindicated. Perhaps we are also within reach of the design of experiments to validate and confront Quantum Thermodynamics versus Quantum Statistical Mechanics in situations where they entail different predictions. 


This website, permanently "under construction provides, below, a selection of references by members of the Keenan school of thermodynamics, as well as a few related references not easily accessible online. We originally meant to collect here some pioneering contributions on the following subjects: 

thermodynamics of irreversible processes - chemical equilibrium - Onsager reciprocity relations -  quantum thermodynamics - foundations of quantum mechanics - microscopic significance of entropy - microscopic origin of irreversibility - uniting mechanics and thermodynamics - general and unambiguous definitions of entropy for nonequilibrium states - maximum entropy production principle (MEPP) - theories of nonequilibrium - ontic status of the density operator -  etc.

However, the field is growing so fast that there is no hope to give here a fair representation. In fact, in this field, because thermodynamics pervades such a broad spectrum of applications, it is very easy to overlook the existing literature. For example, in the recent 'review' paper on MEPP: 

L.M. Martyushev and V.D. Seleznev, Maximum entropy production principle in physics, chemistry, and biology, Physics Reports, 426, 1-45 (2006),

none of the 194 cited references acknowledges our pioneering papers on the quantum version of the subject, perhaps because we called it "steepest entropy ascent ansatz" (see our 1984-86 papers)  instead of "maximum entropy production principle", and so if literature search is done by just googling MEPP our work escapes. Another similar example is the paper

S. Gheorghiu-Svirschevski, Nonlinear quantum evolution with maximal entropy production, Physical Review A, 63, 022105 (2001)

where the nonlinear steepest entropy ascent quantum dynamics we proposed in 1981, was partially "rediscovered" without any citation of our original work, as the author herself later acknowledged in the addendum

S. Gheorghiu-Svirschevski, Addendum to "Nonlinear quantum evolution with maximal entropy production", Physical Review A, 63, 054102 (2001). PDF  

Well after this webpage has been established in 2004, a large group of scientists has tackled the subject and adopted the term Quantum Thermodynamics with a broader and more orthodox meaning than our original use of this term. The goal of this large network of researchers from many different countries, mainly in Europe, is "to significantly advance the theory of foundational thermodynamics, with a focus on its applicability in the nanoscale regime, and to pursue quantum thermodynamics experiments." Researchers in this network have recently published innumerable articles on many quantum thermodynamics related topics. This page will not attempt to follow and catch up with such fast growing literature. The reader be advised that therefore the present webpage will follow only the literature strictly related to the pioneering works on the subject related to the idea of steepest entropy ascent dynamics of nonequilibrium as described for example in the recent paper.


For suggestions please write to

gianpaolo . beretta >at< unibs . it


The following powerpoint presentation and long paper under continuous revisionprovide an introductory compendium about what we meant by Quantum Thermodynamics in te 1980's, the unified theory of mechanics and thermodynamics developed by Hatsopoulos, Gyftopoulos, Park and Beretta, a long standing and ongoing effort to disclose the general microscopic foundations of entropy and irreversibility that drive all natural physical phenomena.

On the 'ontic' interpretation/conjecture of a quantum thermodynamics with intrinsic entropy and intrinsic irreversibility see the online lecture given at the Perimeter Institute, Waterloo, Canada, on October 1, 2009, pirsa.org/09100088/ PDF VIDEO


What is Quantum Thermodynamics? PDF


List of References

Below is a selection of references by members of the Keenan school of thermodynamics, as well as a few related references not easily accessible online: 

Webpage established in 2004.


Google Scholar     



click over cover image for details